0:00:03 > 0:00:07Throughout history, humankind has struggled
0:00:07 > 0:00:11to understand the fundamental workings of the material world.
0:00:11 > 0:00:16We've endeavoured to discover the rules and patterns that determine the qualities
0:00:16 > 0:00:22of the objects that surround us, and their complex relationship to us and each other.
0:00:23 > 0:00:28Over thousands of years, societies all over the world have found that one discipline
0:00:28 > 0:00:31above all others yields certain knowledge
0:00:31 > 0:00:35about the underlying realities of the physical world.
0:00:35 > 0:00:38That discipline is mathematics.
0:00:38 > 0:00:41I'm Marcus Du Sautoy, and I'm a mathematician.
0:00:41 > 0:00:46I see myself as a pattern searcher, hunting down the hidden structures
0:00:46 > 0:00:51that lie behind the apparent chaos and complexity of the world around us.
0:00:52 > 0:00:58In my search for pattern and order, I draw upon the work of the great mathematicians
0:00:58 > 0:01:02who've gone before me, people belonging to cultures across the globe,
0:01:02 > 0:01:06whose innovations created the language the universe is written in.
0:01:06 > 0:01:12I want to take you on a journey through time and space, and track the growth of mathematics
0:01:12 > 0:01:16from its awakening to the sophisticated subject we know today.
0:01:18 > 0:01:21Using computer generated imagery, we will explore
0:01:21 > 0:01:24the trailblazing discoveries that allowed the earliest civilisations
0:01:24 > 0:01:28to understand the world mathematical.
0:01:28 > 0:01:31This is the story of maths.
0:01:51 > 0:01:55Our world is made up of patterns and sequences.
0:01:55 > 0:01:57They're all around us.
0:01:57 > 0:01:59Day becomes night.
0:01:59 > 0:02:04Animals travel across the earth in ever-changing formations.
0:02:04 > 0:02:08Landscapes are constantly altering.
0:02:08 > 0:02:12One of the reasons mathematics began was because we needed to find a way
0:02:12 > 0:02:15of making sense of these natural patterns.
0:02:18 > 0:02:23The most basic concepts of maths - space and quantity -
0:02:23 > 0:02:27are hard-wired into our brains.
0:02:27 > 0:02:30Even animals have a sense of distance and number,
0:02:30 > 0:02:36assessing when their pack is outnumbered, and whether to fight or fly,
0:02:36 > 0:02:40calculating whether their prey is within striking distance.
0:02:40 > 0:02:46Understanding maths is the difference between life and death.
0:02:47 > 0:02:50But it was man who took these basic concepts
0:02:50 > 0:02:52and started to build upon these foundations.
0:02:52 > 0:02:55At some point, humans started to spot patterns,
0:02:55 > 0:02:59to make connections, to count and to order the world around them.
0:02:59 > 0:03:04With this, a whole new mathematical universe began to emerge.
0:03:11 > 0:03:12This is the River Nile.
0:03:12 > 0:03:15It's been the lifeline of Egypt for millennia.
0:03:17 > 0:03:20I've come here because it's where some of the first signs
0:03:20 > 0:03:23of mathematics as we know it today emerged.
0:03:25 > 0:03:30People abandoned nomadic life and began settling here as early as 6000BC.
0:03:30 > 0:03:34The conditions were perfect for farming.
0:03:38 > 0:03:44The most important event for Egyptian agriculture each year was the flooding of the Nile.
0:03:44 > 0:03:49So this was used as a marker to start each new year.
0:03:49 > 0:03:54Egyptians did record what was going on over periods of time,
0:03:54 > 0:03:56so in order to establish a calendar like this,
0:03:56 > 0:03:59you need to count how many days, for example,
0:03:59 > 0:04:02happened in-between lunar phases,
0:04:02 > 0:04:08or how many days happened in-between two floodings of the Nile.
0:04:10 > 0:04:14Recording the patterns for the seasons was essential,
0:04:14 > 0:04:18not only to their management of the land, but also their religion.
0:04:18 > 0:04:21The ancient Egyptians who settled on the Nile banks
0:04:21 > 0:04:25believed it was the river god, Hapy, who flooded the river each year.
0:04:25 > 0:04:28And in return for the life-giving water,
0:04:28 > 0:04:32the citizens offered a portion of the yield as a thanksgiving.
0:04:34 > 0:04:38As settlements grew larger, it became necessary to find ways to administer them.
0:04:38 > 0:04:43Areas of land needed to be calculated, crop yields predicted,
0:04:43 > 0:04:45taxes charged and collated.
0:04:45 > 0:04:49In short, people needed to count and measure.
0:04:50 > 0:04:53The Egyptians used their bodies to measure the world,
0:04:53 > 0:04:56and it's how their units of measurements evolved.
0:04:56 > 0:04:59A palm was the width of a hand,
0:04:59 > 0:05:03a cubit an arm length from elbow to fingertips.
0:05:03 > 0:05:07Land cubits, strips of land measuring a cubit by 100,
0:05:07 > 0:05:10were used by the pharaoh's surveyors to calculate areas.
0:05:13 > 0:05:17There's a very strong link between bureaucracy
0:05:17 > 0:05:20and the development of mathematics in ancient Egypt.
0:05:20 > 0:05:23And we can see this link right from the beginning,
0:05:23 > 0:05:25from the invention of the number system,
0:05:25 > 0:05:28throughout Egyptian history, really.
0:05:28 > 0:05:30For the Old Kingdom, the only evidence we have
0:05:30 > 0:05:34are metrological systems, that is measurements for areas, for length.
0:05:34 > 0:05:41This points to a bureaucratic need to develop such things.
0:05:41 > 0:05:46It was vital to know the area of a farmer's land so he could be taxed accordingly.
0:05:46 > 0:05:51Or if the Nile robbed him of part of his land, so he could request a rebate.
0:05:51 > 0:05:54It meant that the pharaoh's surveyors were often calculating
0:05:54 > 0:05:58the area of irregular parcels of land.
0:05:58 > 0:06:00It was the need to solve such practical problems
0:06:00 > 0:06:05that made them the earliest mathematical innovators.
0:06:09 > 0:06:13The Egyptians needed some way to record the results of their calculations.
0:06:15 > 0:06:20Amongst all the hieroglyphs that cover the tourist souvenirs scattered around Cairo,
0:06:20 > 0:06:25I was on the hunt for those that recorded some of the first numbers in history.
0:06:25 > 0:06:29They were difficult to track down.
0:06:30 > 0:06:33But I did find them in the end.
0:06:36 > 0:06:41The Egyptians were using a decimal system, motivated by the 10 fingers on our hands.
0:06:41 > 0:06:44The sign for one was a stroke,
0:06:44 > 0:06:5010, a heel bone, 100, a coil of rope, and 1,000, a Lotus plant.
0:06:50 > 0:06:52How much is this T-shirt?
0:06:52 > 0:06:54Er, 25.
0:06:54 > 0:07:00- 25!- Yes!- So that would be 2 knee bones and 5 strokes.
0:07:00 > 0:07:03- So you're not gonna charge me anything up here?- Here, one million!
0:07:03 > 0:07:05- One million?- My God!
0:07:05 > 0:07:07This one million.
0:07:07 > 0:07:09One million, yeah, that's pretty big!
0:07:11 > 0:07:16The hieroglyphs are beautiful, but the Egyptian number system was fundamentally flawed.
0:07:18 > 0:07:21They had no concept of a place value,
0:07:21 > 0:07:24so one stroke could only represent one unit,
0:07:24 > 0:07:26not 100 or 1,000.
0:07:26 > 0:07:29Although you can write a million with just one character,
0:07:29 > 0:07:33rather than the seven that we use, if you want to write a million minus one,
0:07:33 > 0:07:36then the poor old Egyptian scribe has got to write nine strokes,
0:07:36 > 0:07:40nine heel bones, nine coils of rope, and so on,
0:07:40 > 0:07:42a total of 54 characters.
0:07:44 > 0:07:50Despite the drawback of this number system, the Egyptians were brilliant problem solvers.
0:07:52 > 0:07:56We know this because of the few records that have survived.
0:07:56 > 0:07:59The Egyptian scribes used sheets of papyrus
0:07:59 > 0:08:02to record their mathematical discoveries.
0:08:02 > 0:08:06This delicate material made from reeds decayed over time
0:08:06 > 0:08:09and many secrets perished with it.
0:08:09 > 0:08:13But there's one revealing document that has survived.
0:08:13 > 0:08:17The Rhind Mathematical Papyrus is the most important document
0:08:17 > 0:08:20we have today for Egyptian mathematics.
0:08:20 > 0:08:24We get a good overview of what types of problems
0:08:24 > 0:08:28the Egyptians would have dealt with in their mathematics.
0:08:28 > 0:08:34We also get explicitly stated how multiplications and divisions were carried out.
0:08:35 > 0:08:40The papyri show how to multiply two large numbers together.
0:08:40 > 0:08:44But to illustrate the method, let's take two smaller numbers.
0:08:44 > 0:08:47Let's do three times six.
0:08:47 > 0:08:50The scribe would take the first number, three, and put it in one column.
0:08:53 > 0:08:56In the second column, he would place the number one.
0:08:56 > 0:09:00Then he would double the numbers in each column, so three becomes six...
0:09:04 > 0:09:06..and six would become 12.
0:09:11 > 0:09:14And then in the second column, one would become two,
0:09:14 > 0:09:16and two becomes four.
0:09:18 > 0:09:21Now, here's the really clever bit.
0:09:21 > 0:09:24The scribe wants to multiply three by six.
0:09:24 > 0:09:27So he takes the powers of two in the second column,
0:09:27 > 0:09:31which add up to six. That's two plus four.
0:09:31 > 0:09:34Then he moves back to the first column, and just takes
0:09:34 > 0:09:37those rows corresponding to the two and the four.
0:09:37 > 0:09:39So that's six and the 12.
0:09:39 > 0:09:43He adds those together to get the answer of 18.
0:09:43 > 0:09:47But for me, the most striking thing about this method
0:09:47 > 0:09:51is that the scribe has effectively written that second number in binary.
0:09:51 > 0:09:56Six is one lot of four, one lot of two, and no units.
0:09:56 > 0:09:59Which is 1-1-0.
0:09:59 > 0:10:03The Egyptians have understood the power of binary over 3,000 years
0:10:03 > 0:10:07before the mathematician and philosopher Leibniz would reveal their potential.
0:10:07 > 0:10:11Today, the whole technological world depends on the same principles
0:10:11 > 0:10:14that were used in ancient Egypt.
0:10:16 > 0:10:22The Rhind Papyrus was recorded by a scribe called Ahmes around 1650BC.
0:10:22 > 0:10:27Its problems are concerned with finding solutions to everyday situations.
0:10:27 > 0:10:30Several of the problems mention bread and beer,
0:10:30 > 0:10:33which isn't surprising as Egyptian workers were paid in food and drink.
0:10:33 > 0:10:37One is concerned with how to divide nine loaves
0:10:37 > 0:10:41equally between 10 people, without a fight breaking out.
0:10:41 > 0:10:44I've got nine loaves of bread here.
0:10:44 > 0:10:47I'm gonna take five of them and cut them into halves.
0:10:48 > 0:10:51Of course, nine people could shave a 10th off their loaf
0:10:51 > 0:10:54and give the pile of crumbs to the 10th person.
0:10:54 > 0:10:58But the Egyptians developed a far more elegant solution -
0:10:58 > 0:11:02take the next four and divide those into thirds.
0:11:04 > 0:11:07But two of the thirds I am now going to cut into fifths,
0:11:07 > 0:11:10so each piece will be one fifteenth.
0:11:12 > 0:11:17Each person then gets one half
0:11:17 > 0:11:19and one third
0:11:19 > 0:11:23and one fifteenth.
0:11:23 > 0:11:26It is through such seemingly practical problems
0:11:26 > 0:11:29that we start to see a more abstract mathematics developing.
0:11:29 > 0:11:32Suddenly, new numbers are on the scene - fractions -
0:11:32 > 0:11:37and it isn't too long before the Egyptians are exploring the mathematics of these numbers.
0:11:39 > 0:11:45Fractions are clearly of practical importance to anyone dividing quantities for trade in the market.
0:11:45 > 0:11:51To log these transactions, the Egyptians developed notation which recorded these new numbers.
0:11:53 > 0:11:56One of the earliest representations of these fractions
0:11:56 > 0:12:00came from a hieroglyph which had great mystical significance.
0:12:00 > 0:12:03It's called the Eye of Horus.
0:12:03 > 0:12:09Horus was an Old Kingdom god, depicted as half man, half falcon.
0:12:10 > 0:12:15According to legend, Horus' father was killed by his other son, Seth.
0:12:15 > 0:12:18Horus was determined to avenge the murder.
0:12:18 > 0:12:21During one particularly fierce battle,
0:12:21 > 0:12:26Seth ripped out Horus' eye, tore it up and scattered it over Egypt.
0:12:26 > 0:12:29But the gods were looking favourably on Horus.
0:12:29 > 0:12:33They gathered up the scattered pieces and reassembled the eye.
0:12:36 > 0:12:40Each part of the eye represented a different fraction.
0:12:40 > 0:12:43Each one, half the fraction before.
0:12:43 > 0:12:46Although the original eye represented a whole unit,
0:12:46 > 0:12:50the reassembled eye is 1/64 short.
0:12:50 > 0:12:54Although the Egyptians stopped at 1/64,
0:12:54 > 0:12:56implicit in this picture
0:12:56 > 0:12:59is the possibility of adding more fractions,
0:12:59 > 0:13:04halving them each time, the sum getting closer and closer to one,
0:13:04 > 0:13:07but never quite reaching it.
0:13:07 > 0:13:11This is the first hint of something called a geometric series,
0:13:11 > 0:13:14and it appears at a number of points in the Rhind Papyrus.
0:13:14 > 0:13:17But the concept of infinite series would remain hidden
0:13:17 > 0:13:21until the mathematicians of Asia discovered it centuries later.
0:13:24 > 0:13:29Having worked out a system of numbers, including these new fractions,
0:13:29 > 0:13:31it was time for the Egyptians to apply their knowledge
0:13:31 > 0:13:35to understanding shapes that they encountered day to day.
0:13:35 > 0:13:39These shapes were rarely regular squares or rectangles,
0:13:39 > 0:13:44and in the Rhind Papyrus, we find the area of a more organic form, the circle.
0:13:44 > 0:13:48What is astounding in the calculation
0:13:48 > 0:13:51of the area of the circle is its exactness, really.
0:13:51 > 0:13:55How they would have found their method is open to speculation,
0:13:55 > 0:13:57because the texts we have
0:13:57 > 0:14:01do not show us the methods how they were found.
0:14:01 > 0:14:05This calculation is particularly striking because it depends
0:14:05 > 0:14:07on seeing how the shape of the circle
0:14:07 > 0:14:12can be approximated by shapes that the Egyptians already understood.
0:14:12 > 0:14:15The Rhind Papyrus states that a circular field
0:14:15 > 0:14:17with a diameter of nine units
0:14:17 > 0:14:21is close in area to a square with sides of eight.
0:14:21 > 0:14:25But how would this relationship have been discovered?
0:14:25 > 0:14:30My favourite theory sees the answer in the ancient game of mancala.
0:14:30 > 0:14:34Mancala boards were found carved on the roofs of temples.
0:14:34 > 0:14:38Each player starts with an equal number of stones,
0:14:38 > 0:14:41and the object of the game is to move them round the board,
0:14:41 > 0:14:44capturing your opponent's counters on the way.
0:14:45 > 0:14:49As the players sat around waiting to make their next move,
0:14:49 > 0:14:52perhaps one of them realised that sometimes the balls fill the circular holes
0:14:52 > 0:14:54of the mancala board in a rather nice way.
0:14:54 > 0:14:59He might have gone on to experiment with trying to make larger circles.
0:14:59 > 0:15:04Perhaps he noticed that 64 stones, the square of 8,
0:15:04 > 0:15:08can be used to make a circle with diameter nine stones.
0:15:08 > 0:15:13By rearranging the stones, the circle has been approximated by a square.
0:15:13 > 0:15:16And because the area of a circle is pi times the radius squared,
0:15:16 > 0:15:21the Egyptian calculation gives us the first accurate value for pi.
0:15:21 > 0:15:26The area of the circle is 64. Divide this by the radius squared,
0:15:26 > 0:15:30in this case 4.5 squared, and you get a value for pi.
0:15:30 > 0:15:35So 64 divided by 4.5 squared is 3.16,
0:15:35 > 0:15:38just a little under two hundredths away from its true value.
0:15:38 > 0:15:42But the really brilliant thing is, the Egyptians
0:15:42 > 0:15:45are using these smaller shapes to capture the larger shape.
0:15:49 > 0:15:52But there's one imposing and majestic symbol of Egyptian
0:15:52 > 0:15:55mathematics we haven't attempted to unravel yet -
0:15:55 > 0:15:58the pyramid.
0:15:58 > 0:16:03I've seen so many pictures that I couldn't believe I'd be impressed by them.
0:16:03 > 0:16:06But meeting them face to face, you understand why they're called
0:16:06 > 0:16:09one of the Seven Wonders of the Ancient World.
0:16:09 > 0:16:11They're simply breathtaking.
0:16:11 > 0:16:14And how much more impressive they must have been in their day,
0:16:14 > 0:16:19when the sides were as smooth as glass, reflecting the desert sun.
0:16:19 > 0:16:25To me it looks like there might be mirror pyramids hiding underneath the desert,
0:16:25 > 0:16:29which would complete the shapes to make perfectly symmetrical octahedrons.
0:16:29 > 0:16:34Sometimes, in the shimmer of the desert heat, you can almost see these shapes.
0:16:36 > 0:16:43It's the hint of symmetry hidden inside these shapes that makes them so impressive for a mathematician.
0:16:43 > 0:16:47The pyramids are just a little short to create these perfect shapes,
0:16:47 > 0:16:51but some have suggested another important mathematical concept
0:16:51 > 0:16:57might be hidden inside the proportions of the Great Pyramid - the golden ratio.
0:16:57 > 0:17:01Two lengths are in the golden ratio, if the relationship of the longest
0:17:01 > 0:17:07to the shortest is the same as the sum of the two to the longest side.
0:17:07 > 0:17:11Such a ratio has been associated with the perfect proportions one finds
0:17:11 > 0:17:15all over the natural world, as well as in the work of artists,
0:17:15 > 0:17:18architects and designers for millennia.
0:17:22 > 0:17:27Whether the architects of the pyramids were conscious of this important mathematical idea,
0:17:27 > 0:17:32or were instinctively drawn to it because of its satisfying aesthetic properties, we'll never know.
0:17:32 > 0:17:37For me, the most impressive thing about the pyramids is the mathematical brilliance
0:17:37 > 0:17:40that went into making them, including the first inkling
0:17:40 > 0:17:44of one of the great theorems of the ancient world, Pythagoras' theorem.
0:17:46 > 0:17:49In order to get perfect right-angled corners on their buildings
0:17:49 > 0:17:54and pyramids, the Egyptians would have used a rope with knots tied in it.
0:17:54 > 0:17:58At some point, the Egyptians realised that if they took a triangle with sides
0:17:58 > 0:18:05marked with three knots, four knots and five knots, it guaranteed them a perfect right-angle.
0:18:05 > 0:18:10This is because three squared, plus four squared, is equal to five squared.
0:18:10 > 0:18:12So we've got a perfect Pythagorean triangle.
0:18:15 > 0:18:20In fact any triangle whose sides satisfy this relationship will give me an 90-degree angle.
0:18:20 > 0:18:23But I'm pretty sure that the Egyptians hadn't got
0:18:23 > 0:18:28this sweeping generalisation of their 3, 4, 5 triangle.
0:18:28 > 0:18:32We would not expect to find the general proof
0:18:32 > 0:18:35because this is not the style of Egyptian mathematics.
0:18:35 > 0:18:39Every problem was solved using concrete numbers and then
0:18:39 > 0:18:43if a verification would be carried out at the end, it would use the result
0:18:43 > 0:18:45and these concrete, given numbers,
0:18:45 > 0:18:49there's no general proof within the Egyptian mathematical texts.
0:18:50 > 0:18:54It would be some 2,000 years before the Greeks and Pythagoras
0:18:54 > 0:18:59would prove that all right-angled triangles shared certain properties.
0:18:59 > 0:19:03This wasn't the only mathematical idea that the Egyptians would anticipate.
0:19:03 > 0:19:10In a 4,000-year-old document called the Moscow papyrus, we find a formula for the volume
0:19:10 > 0:19:16of a pyramid with its peak sliced off, which shows the first hint of calculus at work.
0:19:16 > 0:19:22For a culture like Egypt that is famous for its pyramids, you would expect problems like this
0:19:22 > 0:19:26to have been a regular feature within the mathematical texts.
0:19:26 > 0:19:31The calculation of the volume of a truncated pyramid is one of the most
0:19:31 > 0:19:36advanced bits, according to our modern standards of mathematics,
0:19:36 > 0:19:39that we have from ancient Egypt.
0:19:39 > 0:19:43The architects and engineers would certainly have wanted such a formula
0:19:43 > 0:19:46to calculate the amount of materials required to build it.
0:19:46 > 0:19:49But it's a mark of the sophistication
0:19:49 > 0:19:53of Egyptian mathematics that they were able to produce such a beautiful method.
0:19:59 > 0:20:03To understand how they derived their formula, start with a pyramid
0:20:03 > 0:20:08built such that the highest point sits directly over one corner.
0:20:08 > 0:20:13Three of these can be put together to make a rectangular box,
0:20:13 > 0:20:18so the volume of the skewed pyramid is a third the volume of the box.
0:20:18 > 0:20:24That is, the height, times the length, times the width, divided by three.
0:20:24 > 0:20:29Now comes an argument which shows the very first hints of the calculus at work,
0:20:29 > 0:20:35thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory.
0:20:35 > 0:20:39Suppose you could cut the pyramid into slices, you could then slide
0:20:39 > 0:20:44the layers across to make the more symmetrical pyramid you see in Giza.
0:20:44 > 0:20:49However, the volume of the pyramid has not changed, despite the rearrangement of the layers.
0:20:49 > 0:20:52So the same formula works.
0:20:55 > 0:20:58The Egyptians were amazing innovators,
0:20:58 > 0:21:02and their ability to generate new mathematics was staggering.
0:21:02 > 0:21:07For me, they revealed the power of geometry and numbers, and made the first moves
0:21:07 > 0:21:11towards some of the exciting mathematical discoveries to come.
0:21:11 > 0:21:15But there was another civilisation that had mathematics to rival that of Egypt.
0:21:15 > 0:21:20And we know much more about their achievements.
0:21:24 > 0:21:27This is Damascus, over 5,000 years old,
0:21:27 > 0:21:31and still vibrant and bustling today.
0:21:31 > 0:21:36It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt.
0:21:36 > 0:21:43The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC.
0:21:43 > 0:21:51In order to expand and run their empire, they became masters of managing and manipulating numbers.
0:21:51 > 0:21:53We have law codes for instance that tell us
0:21:53 > 0:21:56about the way society is ordered.
0:21:56 > 0:22:00The people we know most about are the scribes, the professionally literate
0:22:00 > 0:22:05and numerate people who kept the records for the wealthy families and for the temples and palaces.
0:22:05 > 0:22:10Scribe schools existed from around 2500BC.
0:22:10 > 0:22:17Aspiring scribes were sent there as children, and learned how to read, write and work with numbers.
0:22:17 > 0:22:20Scribe records were kept on clay tablets,
0:22:20 > 0:22:24which allowed the Babylonians to manage and advance their empire.
0:22:24 > 0:22:31However, many of the tablets we have today aren't official documents, but children's exercises.
0:22:31 > 0:22:37It's these unlikely relics that give us a rare insight into how the Babylonians approached mathematics.
0:22:37 > 0:22:42So, this is a geometrical textbook from about the 18th century BC.
0:22:42 > 0:22:44I hope you can see that there are lots of pictures on it.
0:22:44 > 0:22:49And underneath each picture is a text that sets a problem about the picture.
0:22:49 > 0:22:55So for instance this one here says, I drew a square, 60 units long,
0:22:55 > 0:23:01and inside it, I drew four circles - what are their areas?
0:23:01 > 0:23:07This little tablet here was written 1,000 years at least later than the tablet here,
0:23:07 > 0:23:10but has a very interesting relationship.
0:23:10 > 0:23:12It also has four circles on,
0:23:12 > 0:23:17in a square, roughly drawn, but this isn't a textbook, it's a school exercise.
0:23:17 > 0:23:21The adult scribe who's teaching the student is being given this
0:23:21 > 0:23:25as an example of completed homework or something like that.
0:23:26 > 0:23:29Like the Egyptians, the Babylonians appeared interested
0:23:29 > 0:23:32in solving practical problems to do with measuring and weighing.
0:23:32 > 0:23:37The Babylonian solutions to these problems are written like mathematical recipes.
0:23:37 > 0:23:43A scribe would simply follow and record a set of instructions to get a result.
0:23:43 > 0:23:47Here's an example of the kind of problem they'd solve.
0:23:47 > 0:23:51I've got a bundle of cinnamon sticks here, but I'm not gonna weigh them.
0:23:51 > 0:23:56Instead, I'm gonna take four times their weight and add them to the scales.
0:23:58 > 0:24:04Now I'm gonna add 20 gin. Gin was the ancient Babylonian measure of weight.
0:24:04 > 0:24:07I'm gonna take half of everything here and then add it again...
0:24:07 > 0:24:10That's two bundles, and ten gin.
0:24:10 > 0:24:16Everything on this side is equal to one mana. One mana was 60 gin.
0:24:16 > 0:24:20And here, we have one of the first mathematical equations in history,
0:24:20 > 0:24:23everything on this side is equal to one mana.
0:24:23 > 0:24:26But how much does the bundle of cinnamon sticks weigh?
0:24:26 > 0:24:29Without any algebraic language, they were able to manipulate
0:24:29 > 0:24:35the quantities to be able to prove that the cinnamon sticks weighed five gin.
0:24:35 > 0:24:40In my mind, it's this kind of problem which gives mathematics a bit of a bad name.
0:24:40 > 0:24:45You can blame those ancient Babylonians for all those tortuous problems you had at school.
0:24:45 > 0:24:50But the ancient Babylonian scribes excelled at this kind of problem.
0:24:50 > 0:24:57Intriguingly, they weren't using powers of 10, like the Egyptians, they were using powers of 60.
0:25:00 > 0:25:05The Babylonians invented their number system, like the Egyptians, by using their fingers.
0:25:05 > 0:25:08But instead of counting through the 10 fingers on their hand,
0:25:08 > 0:25:11Babylonians found a more intriguing way to count body parts.
0:25:11 > 0:25:14They used the 12 knuckles on one hand,
0:25:14 > 0:25:16and the five fingers on the other to be able to count
0:25:16 > 0:25:2012 times 5, ie 60 different numbers.
0:25:20 > 0:25:25So for example, this number would have been 2 lots of 12, 24,
0:25:25 > 0:25:29and then, 1, 2, 3, 4, 5, to make 29.
0:25:32 > 0:25:35The number 60 had another powerful property.
0:25:35 > 0:25:39It can be perfectly divided in a multitude of ways.
0:25:39 > 0:25:41Here are 60 beans.
0:25:41 > 0:25:44I can arrange them in 2 rows of 30.
0:25:48 > 0:25:513 rows of 20.
0:25:51 > 0:25:534 rows of 15.
0:25:53 > 0:25:565 rows of 12.
0:25:56 > 0:25:59Or 6 rows of 10.
0:25:59 > 0:26:04The divisibility of 60 makes it a perfect base in which to do arithmetic.
0:26:04 > 0:26:11The base 60 system was so successful, we still use elements of it today.
0:26:11 > 0:26:15Every time we want to tell the time, we recognise units of 60 -
0:26:15 > 0:26:1960 seconds in a minute, 60 minutes in an hour.
0:26:19 > 0:26:24But the most important feature of the Babylonians' number system was that it recognised place value.
0:26:24 > 0:26:30Just as our decimal numbers count how many lots of tens, hundreds and thousands you're recording,
0:26:30 > 0:26:34the position of each Babylonian number records the power of 60.
0:26:41 > 0:26:44Instead of inventing new symbols for bigger and bigger numbers,
0:26:44 > 0:26:50they would write 1-1-1, so this number would be 3,661.
0:26:54 > 0:26:59The catalyst for this discovery was the Babylonians' desire to chart the course of the night sky.
0:27:07 > 0:27:10The Babylonians' calendar was based on the cycles of the moon.
0:27:10 > 0:27:15They needed a way of recording astronomically large numbers.
0:27:15 > 0:27:19Month by month, year by year, these cycles were recorded.
0:27:19 > 0:27:25From about 800BC, there were complete lists of lunar eclipses.
0:27:25 > 0:27:30The Babylonian system of measurement was quite sophisticated at that time.
0:27:30 > 0:27:32They had a system of angular measurement,
0:27:32 > 0:27:36360 degrees in a full circle, each degree was divided
0:27:36 > 0:27:41into 60 minutes, a minute was further divided into 60 seconds.
0:27:41 > 0:27:48So they had a regular system for measurement, and it was in perfect harmony with their number system,
0:27:48 > 0:27:52so it's well suited not only for observation but also for calculation.
0:27:52 > 0:27:56But in order to calculate and cope with these large numbers,
0:27:56 > 0:28:00the Babylonians needed to invent a new symbol.
0:28:00 > 0:28:03And in so doing, they prepared the ground for one of the great
0:28:03 > 0:28:06breakthroughs in the history of mathematics - zero.
0:28:06 > 0:28:11In the early days, the Babylonians, in order to mark an empty place in
0:28:11 > 0:28:14the middle of a number, would simply leave a blank space.
0:28:14 > 0:28:19So they needed a way of representing nothing in the middle of a number.
0:28:19 > 0:28:25So they used a sign, as a sort of breathing marker, a punctuation mark,
0:28:25 > 0:28:28and it comes to mean zero in the middle of a number.
0:28:28 > 0:28:31This was the first time zero, in any form,
0:28:31 > 0:28:35had appeared in the mathematical universe.
0:28:35 > 0:28:42But it would be over a 1,000 years before this little place holder would become a number in its own right.
0:28:50 > 0:28:53Having established such a sophisticated system of numbers,
0:28:53 > 0:28:59they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia.
0:29:02 > 0:29:06Babylonian engineers and surveyors found ingenious ways of
0:29:06 > 0:29:10accessing water, and channelling it to the crop fields.
0:29:10 > 0:29:15Yet again, they used mathematics to come up with solutions.
0:29:15 > 0:29:19The Orontes valley in Syria is still an agricultural hub,
0:29:19 > 0:29:26and the old methods of irrigation are being exploited today, just as they were thousands of years ago.
0:29:26 > 0:29:29Many of the problems in Babylonian mathematics
0:29:29 > 0:29:34are concerned with measuring land, and it's here we see for the first time
0:29:34 > 0:29:39the use of quadratic equations, one of the greatest legacies of Babylonian mathematics.
0:29:39 > 0:29:43Quadratic equations involve things where the unknown quantity
0:29:43 > 0:29:46you're trying to identify is multiplied by itself.
0:29:46 > 0:29:49We call this squaring because it gives the area of a square,
0:29:49 > 0:29:53and it's in the context of calculating the area of land
0:29:53 > 0:29:55that these quadratic equations naturally arise.
0:30:01 > 0:30:03Here's a typical problem.
0:30:03 > 0:30:06If a field has an area of 55 units
0:30:06 > 0:30:10and one side is six units longer than the other,
0:30:10 > 0:30:12how long is the shorter side?
0:30:14 > 0:30:18The Babylonian solution was to reconfigure the field as a square.
0:30:18 > 0:30:21Cut three units off the end
0:30:21 > 0:30:24and move this round.
0:30:24 > 0:30:29Now, there's a three-by-three piece missing, so let's add this in.
0:30:29 > 0:30:34The area of the field has increased by nine units.
0:30:34 > 0:30:38This makes the new area 64.
0:30:38 > 0:30:41So the sides of the square are eight units.
0:30:41 > 0:30:45The problem-solver knows that they've added three to this side.
0:30:45 > 0:30:49So, the original length must be five.
0:30:50 > 0:30:55It may not look like it, but this is one of the first quadratic equations in history.
0:30:57 > 0:31:02In modern mathematics, I would use the symbolic language of algebra to solve this problem.
0:31:02 > 0:31:07The amazing feat of the Babylonians is that they were using these geometric games to find the value,
0:31:07 > 0:31:10without any recourse to symbols or formulas.
0:31:10 > 0:31:13The Babylonians were enjoying problem-solving for its own sake.
0:31:13 > 0:31:17They were falling in love with mathematics.
0:31:29 > 0:31:34The Babylonians' fascination with numbers soon found a place in their leisure time, too.
0:31:34 > 0:31:35They were avid game-players.
0:31:35 > 0:31:38The Babylonians and their descendants have been playing
0:31:38 > 0:31:43a version of backgammon for over 5,000 years.
0:31:43 > 0:31:45The Babylonians played board games,
0:31:45 > 0:31:52from very posh board games in royal tombs to little bits of board games found in schools,
0:31:52 > 0:31:56to board games scratched on the entrances of palaces,
0:31:56 > 0:32:00so that the guardsmen must have played when they were bored,
0:32:00 > 0:32:03and they used dice to move their counters round.
0:32:04 > 0:32:09People who played games were using numbers in their leisure time to try and outwit their opponent,
0:32:09 > 0:32:12doing mental arithmetic very fast,
0:32:12 > 0:32:17and so they were calculating in their leisure time,
0:32:17 > 0:32:21without even thinking about it as being mathematical hard work.
0:32:23 > 0:32:24Now's my chance.
0:32:24 > 0:32:30'I hadn't played backgammon for ages but I reckoned my maths would give me a fighting chance.'
0:32:30 > 0:32:33- It's up to you.- Six... I need to move something.
0:32:33 > 0:32:36'But it wasn't as easy as I thought.'
0:32:36 > 0:32:38Ah! What the hell was that?
0:32:38 > 0:32:42- Yeah.- This is one, this is two.
0:32:42 > 0:32:44Now you're in trouble.
0:32:44 > 0:32:47- So I can't move anything. - You cannot move these.
0:32:47 > 0:32:49Oh, gosh.
0:32:50 > 0:32:52There you go.
0:32:53 > 0:32:54Three and four.
0:32:54 > 0:33:00'Just like the ancient Babylonians, my opponents were masters of tactical mathematics.'
0:33:00 > 0:33:02Yeah.
0:33:03 > 0:33:05Put it there. Good game.
0:33:07 > 0:33:10The Babylonians are recognised as one of the first cultures
0:33:10 > 0:33:13to use symmetrical mathematical shapes to make dice,
0:33:13 > 0:33:17but there is more heated debate about whether they might also
0:33:17 > 0:33:20have been the first to discover the secrets of another important shape.
0:33:20 > 0:33:24The right-angled triangle.
0:33:27 > 0:33:32We've already seen how the Egyptians use a 3-4-5 right-angled triangle.
0:33:32 > 0:33:37But what the Babylonians knew about this shape and others like it is much more sophisticated.
0:33:37 > 0:33:42This is the most famous and controversial ancient tablet we have.
0:33:42 > 0:33:44It's called Plimpton 322.
0:33:45 > 0:33:49Many mathematicians are convinced it shows the Babylonians
0:33:49 > 0:33:53could well have known the principle regarding right-angled triangles,
0:33:53 > 0:33:57that the square on the diagonal is the sum of the squares on the sides,
0:33:57 > 0:34:00and known it centuries before the Greeks claimed it.
0:34:01 > 0:34:06This is a copy of arguably the most famous Babylonian tablet,
0:34:06 > 0:34:08which is Plimpton 322,
0:34:08 > 0:34:12and these numbers here reflect the width or height of a triangle,
0:34:12 > 0:34:17this being the diagonal, the other side would be over here,
0:34:17 > 0:34:19and the square of this column
0:34:19 > 0:34:23plus the square of the number in this column
0:34:23 > 0:34:26equals the square of the diagonal.
0:34:26 > 0:34:31They are arranged in an order of steadily decreasing angle,
0:34:31 > 0:34:34on a very uniform basis, showing that somebody
0:34:34 > 0:34:38had a lot of understanding of how the numbers fit together.
0:34:44 > 0:34:50Here were 15 perfect Pythagorean triangles, all of whose sides had whole-number lengths.
0:34:50 > 0:34:56It's tempting to think that the Babylonians were the first custodians of Pythagoras' theorem,
0:34:56 > 0:35:01and it's a conclusion that generations of historians have been seduced by.
0:35:01 > 0:35:03But there could be a much simpler explanation
0:35:03 > 0:35:07for the sets of three numbers which fulfil Pythagoras' theorem.
0:35:07 > 0:35:12It's not a systematic explanation of Pythagorean triples, it's simply
0:35:12 > 0:35:17a mathematics teacher doing some quite complicated calculations,
0:35:17 > 0:35:21but in order to produce some very simple numbers,
0:35:21 > 0:35:26in order to set his students problems about right-angled triangles,
0:35:26 > 0:35:31and in that sense it's about Pythagorean triples only incidentally.
0:35:33 > 0:35:39The most valuable clues to what they understood could lie elsewhere.
0:35:39 > 0:35:43This small school exercise tablet is nearly 4,000 years old
0:35:43 > 0:35:48and reveals just what the Babylonians did know about right-angled triangles.
0:35:48 > 0:35:54It uses a principle of Pythagoras' theorem to find the value of an astounding new number.
0:35:57 > 0:36:05Drawn along the diagonal is a really very good approximation to the square root of two,
0:36:05 > 0:36:10and so that shows us that it was known and used in school environments.
0:36:10 > 0:36:12Why's this important?
0:36:12 > 0:36:18Because the square root of two is what we now call an irrational number,
0:36:18 > 0:36:23that is, if we write it out in decimals, or even in sexigesimal places,
0:36:23 > 0:36:28it doesn't end, the numbers go on forever after the decimal point.
0:36:29 > 0:36:33The implications of this calculation are far-reaching.
0:36:33 > 0:36:37Firstly, it means the Babylonians knew something of Pythagoras' theorem
0:36:37 > 0:36:391,000 years before Pythagoras.
0:36:39 > 0:36:45Secondly, the fact that they can calculate this number to an accuracy of four decimal places
0:36:45 > 0:36:50shows an amazing arithmetic facility, as well as a passion for mathematical detail.
0:36:52 > 0:36:56The Babylonians' mathematical dexterity was astounding,
0:36:56 > 0:37:03and for nearly 2,000 years they spearheaded intellectual progress in the ancient world.
0:37:03 > 0:37:08But when their imperial power began to wane, so did their intellectual vigour.
0:37:16 > 0:37:23By 330BC, the Greeks had advanced their imperial reach into old Mesopotamia.
0:37:25 > 0:37:31This is Palmyra in central Syria, a once-great city built by the Greeks.
0:37:33 > 0:37:41The mathematical expertise needed to build structures with such geometric perfection is impressive.
0:37:42 > 0:37:48Just like the Babylonians before them, the Greeks were passionate about mathematics.
0:37:50 > 0:37:53The Greeks were clever colonists.
0:37:53 > 0:37:56They took the best from the civilisations they invaded
0:37:56 > 0:37:58to advance their own power and influence,
0:37:58 > 0:38:01but they were soon making contributions themselves.
0:38:01 > 0:38:07In my opinion, their greatest innovation was to do with a shift in the mind.
0:38:07 > 0:38:11What they initiated would influence humanity for centuries.
0:38:11 > 0:38:14They gave us the power of proof.
0:38:14 > 0:38:18Somehow they decided that they had to have a deductive system
0:38:18 > 0:38:19for their mathematics
0:38:19 > 0:38:21and the typical deductive system
0:38:21 > 0:38:25was to begin with certain axioms, which you assume are true.
0:38:25 > 0:38:29It's as if you assume a certain theorem is true without proving it.
0:38:29 > 0:38:34And then, using logical methods and very careful steps,
0:38:34 > 0:38:37from these axioms you prove theorems
0:38:37 > 0:38:42and from those theorems you prove more theorems, and it just snowballs.
0:38:43 > 0:38:47Proof is what gives mathematics its strength.
0:38:47 > 0:38:51It's the power of proof which means that the discoveries of the Greeks
0:38:51 > 0:38:55are as true today as they were 2,000 years ago.
0:38:55 > 0:39:01I needed to head west into the heart of the old Greek empire to learn more.
0:39:08 > 0:39:14For me, Greek mathematics has always been heroic and romantic.
0:39:15 > 0:39:20I'm on my way to Samos, less than a mile from the Turkish coast.
0:39:20 > 0:39:25This place has become synonymous with the birth of Greek mathematics,
0:39:25 > 0:39:27and it's down to the legend of one man.
0:39:31 > 0:39:33His name is Pythagoras.
0:39:33 > 0:39:36The legends that surround his life and work have contributed
0:39:36 > 0:39:40to the celebrity status he has gained over the last 2,000 years.
0:39:40 > 0:39:44He's credited, rightly or wrongly, with beginning the transformation
0:39:44 > 0:39:50from mathematics as a tool for accounting to the analytic subject we recognise today.
0:39:54 > 0:39:57Pythagoras is a controversial figure.
0:39:57 > 0:40:00Because he left no mathematical writings, many have questioned
0:40:00 > 0:40:04whether he indeed solved any of the theorems attributed to him.
0:40:04 > 0:40:07He founded a school in Samos in the sixth century BC,
0:40:07 > 0:40:13but his teachings were considered suspect and the Pythagoreans a bizarre sect.
0:40:14 > 0:40:19There is good evidence that there were schools of Pythagoreans,
0:40:19 > 0:40:22and they may have looked more like sects
0:40:22 > 0:40:25than what we associate with philosophical schools,
0:40:25 > 0:40:30because they didn't just share knowledge, they also shared a way of life.
0:40:30 > 0:40:36There may have been communal living and they all seemed to have been
0:40:36 > 0:40:40involved in the politics of their cities.
0:40:40 > 0:40:45One feature that makes them unusual in the ancient world is that they included women.
0:40:46 > 0:40:52But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians -
0:40:52 > 0:40:56the properties of right-angled triangles.
0:40:56 > 0:40:58What's known as Pythagoras' theorem
0:40:58 > 0:41:01states that if you take any right-angled triangle,
0:41:01 > 0:41:05build squares on all the sides, then the area of the largest square
0:41:05 > 0:41:09is equal to the sum of the squares on the two smaller sides.
0:41:13 > 0:41:16It's at this point for me that mathematics is born
0:41:16 > 0:41:19and a gulf opens up between the other sciences,
0:41:19 > 0:41:24and the proof is as simple as it is devastating in its implications.
0:41:24 > 0:41:28Place four copies of the right-angled triangle
0:41:28 > 0:41:29on top of this surface.
0:41:29 > 0:41:31The square that you now see
0:41:31 > 0:41:35has sides equal to the hypotenuse of the triangle.
0:41:35 > 0:41:37By sliding these triangles around,
0:41:37 > 0:41:40we see how we can break the area of the large square up
0:41:40 > 0:41:43into the sum of two smaller squares,
0:41:43 > 0:41:47whose sides are given by the two short sides of the triangle.
0:41:47 > 0:41:52In other words, the square on the hypotenuse is equal to the sum
0:41:52 > 0:41:55of the squares on the other sides. Pythagoras' theorem.
0:41:58 > 0:42:02It illustrates one of the characteristic themes of Greek mathematics -
0:42:02 > 0:42:07the appeal to beautiful arguments in geometry rather than a reliance on number.
0:42:11 > 0:42:16Pythagoras may have fallen out of favour and many of the discoveries accredited to him
0:42:16 > 0:42:21have been contested recently, but there's one mathematical theory that I'm loath to take away from him.
0:42:21 > 0:42:25It's to do with music and the discovery of the harmonic series.
0:42:27 > 0:42:31The story goes that, walking past a blacksmith's one day,
0:42:31 > 0:42:33Pythagoras heard anvils being struck,
0:42:33 > 0:42:38and noticed how the notes being produced sounded in perfect harmony.
0:42:38 > 0:42:42He believed that there must be some rational explanation
0:42:42 > 0:42:46to make sense of why the notes sounded so appealing.
0:42:46 > 0:42:48The answer was mathematics.
0:42:53 > 0:42:58Experimenting with a stringed instrument, Pythagoras discovered that the intervals between
0:42:58 > 0:43:02harmonious musical notes were always represented as whole-number ratios.
0:43:05 > 0:43:08And here's how he might have constructed his theory.
0:43:10 > 0:43:13First, play a note on the open string.
0:43:13 > 0:43:15MAN PLAYS NOTE
0:43:15 > 0:43:17Next, take half the length.
0:43:18 > 0:43:22The note almost sounds the same as the first note.
0:43:22 > 0:43:27In fact it's an octave higher, but the relationship is so strong, we give these notes the same name.
0:43:27 > 0:43:28Now take a third the length.
0:43:31 > 0:43:35We get another note which sounds harmonious next to the first two,
0:43:35 > 0:43:41but take a length of string which is not in a whole-number ratio and all we get is dissonance.
0:43:46 > 0:43:51According to legend, Pythagoras was so excited by this discovery
0:43:51 > 0:43:54that he concluded the whole universe was built from numbers.
0:43:54 > 0:44:00But he and his followers were in for a rather unsettling challenge to their world view
0:44:00 > 0:44:05and it came about as a result of the theorem which bears Pythagoras' name.
0:44:07 > 0:44:12Legend has it, one of his followers, a mathematician called Hippasus,
0:44:12 > 0:44:15set out to find the length of the diagonal
0:44:15 > 0:44:19for a right-angled triangle with two sides measuring one unit.
0:44:19 > 0:44:25Pythagoras' theorem implied that the length of the diagonal was a number whose square was two.
0:44:25 > 0:44:29The Pythagoreans assumed that the answer would be a fraction,
0:44:29 > 0:44:36but when Hippasus tried to express it in this way, no matter how he tried, he couldn't capture it.
0:44:36 > 0:44:38Eventually he realised his mistake.
0:44:38 > 0:44:43It was the assumption that the value was a fraction at all which was wrong.
0:44:43 > 0:44:49The value of the square root of two was the number that the Babylonians etched into the Yale tablet.
0:44:49 > 0:44:53However, they didn't recognise the special character of this number.
0:44:53 > 0:44:55But Hippasus did.
0:44:55 > 0:44:57It was an irrational number.
0:45:00 > 0:45:04The discovery of this new number, and others like it, is akin to an explorer
0:45:04 > 0:45:09discovering a new continent, or a naturalist finding a new species.
0:45:09 > 0:45:13But these irrational numbers didn't fit the Pythagorean world view.
0:45:13 > 0:45:19Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy,
0:45:19 > 0:45:21but Hippasus let slip the discovery
0:45:21 > 0:45:25and was promptly drowned for his attempts to broadcast their research.
0:45:27 > 0:45:32But these mathematical discoveries could not be easily suppressed.
0:45:32 > 0:45:37Schools of philosophy and science started to flourish all over Greece, building on these foundations.
0:45:37 > 0:45:42The most famous of these was the Academy.
0:45:42 > 0:45:47Plato founded this school in Athens in 387 BC.
0:45:47 > 0:45:54Although we think of him today as a philosopher, he was one of mathematics' most important patrons.
0:45:54 > 0:45:57Plato was enraptured by the Pythagorean world view
0:45:57 > 0:46:02and considered mathematics the bedrock of knowledge.
0:46:02 > 0:46:07Some people would say that Plato is the most influential figure
0:46:07 > 0:46:10for our perception of Greek mathematics.
0:46:10 > 0:46:15He argued that mathematics is an important form of knowledge
0:46:15 > 0:46:17and does have a connection with reality.
0:46:17 > 0:46:23So by knowing mathematics, we know more about reality.
0:46:23 > 0:46:29In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking
0:46:29 > 0:46:33the secrets of the universe, a view still held by scientists today.
0:46:33 > 0:46:37Indeed, the importance Plato attached to geometry is encapsulated
0:46:37 > 0:46:43in the sign that was mounted above the Academy, "Let no-one ignorant of geometry enter here."
0:46:47 > 0:46:53Plato proposed that the universe could be crystallised into five regular symmetrical shapes.
0:46:53 > 0:46:56These shapes, which we now call the Platonic solids,
0:46:56 > 0:46:59were composed of regular polygons, assembled to create
0:46:59 > 0:47:03three-dimensional symmetrical objects.
0:47:03 > 0:47:05The tetrahedron represented fire.
0:47:05 > 0:47:09The icosahedron, made from 20 triangles, represented water.
0:47:09 > 0:47:12The stable cube was Earth.
0:47:12 > 0:47:15The eight-faced octahedron was air.
0:47:15 > 0:47:19And the fifth Platonic solid, the dodecahedron,
0:47:19 > 0:47:22made out of 12 pentagons, was reserved for the shape
0:47:22 > 0:47:26that captured Plato's view of the universe.
0:47:29 > 0:47:33Plato's theory would have a seismic influence and continued to inspire
0:47:33 > 0:47:37mathematicians and astronomers for over 1,500 years.
0:47:38 > 0:47:41In addition to the breakthroughs made in the Academy,
0:47:41 > 0:47:45mathematical triumphs were also emerging from the edge of the Greek empire,
0:47:45 > 0:47:51and owed as much to the mathematical heritage of the Egyptians as the Greeks.
0:47:51 > 0:47:58Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC,
0:47:58 > 0:48:04and its famous library soon gained a reputation to rival Plato's Academy.
0:48:04 > 0:48:11The kings of Alexandria were prepared to invest in the arts and culture,
0:48:11 > 0:48:14in technology, mathematics, grammar,
0:48:14 > 0:48:19because patronage for cultural pursuits
0:48:19 > 0:48:27was one way of showing that you were a more prestigious ruler,
0:48:27 > 0:48:30and had a better entitlement to greatness.
0:48:32 > 0:48:35The old library and its precious contents were destroyed
0:48:35 > 0:48:38But its spirit is alive in a new building.
0:48:40 > 0:48:44Today, the library remains a place of discovery and scholarship.
0:48:48 > 0:48:51Mathematicians and philosophers flocked to Alexandria,
0:48:51 > 0:48:55driven by their thirst for knowledge and the pursuit of excellence.
0:48:55 > 0:48:59The patrons of the library were the first professional scientists,
0:48:59 > 0:49:02individuals who were paid for their devotion to research.
0:49:02 > 0:49:04But of all those early pioneers,
0:49:04 > 0:49:08my hero is the enigmatic Greek mathematician Euclid.
0:49:12 > 0:49:15We know very little about Euclid's life,
0:49:15 > 0:49:19but his greatest achievements were as a chronicler of mathematics.
0:49:19 > 0:49:24Around 300 BC, he wrote the most important text book of all time -
0:49:24 > 0:49:27The Elements. In The Elements,
0:49:27 > 0:49:31we find the culmination of the mathematical revolution
0:49:31 > 0:49:32which had taken place in Greece.
0:49:34 > 0:49:39It's built on a series of mathematical assumptions, called axioms.
0:49:39 > 0:49:44For example, a line can be drawn between any two points.
0:49:44 > 0:49:48From these axioms, logical deductions are made and mathematical theorems established.
0:49:51 > 0:49:56The Elements contains formulas for calculating the volumes of cones
0:49:56 > 0:49:59and cylinders, proofs about geometric series,
0:49:59 > 0:50:02perfect numbers and primes.
0:50:02 > 0:50:06The climax of The Elements is a proof that there are only five Platonic solids.
0:50:09 > 0:50:14For me, this last theorem captures the power of mathematics.
0:50:14 > 0:50:17It's one thing to build five symmetrical solids,
0:50:17 > 0:50:22quite another to come up with a watertight, logical argument for why there can't be a sixth.
0:50:22 > 0:50:26The Elements unfolds like a wonderful, logical mystery novel.
0:50:26 > 0:50:29But this is a story which transcends time.
0:50:29 > 0:50:33Scientific theories get knocked down, from one generation to the next,
0:50:33 > 0:50:39but the theorems in The Elements are as true today as they were 2,000 years ago.
0:50:39 > 0:50:43When you stop and think about it, it's really amazing.
0:50:43 > 0:50:45It's the same theorems that we teach.
0:50:45 > 0:50:49We may teach them in a slightly different way, we may organise them differently,
0:50:49 > 0:50:54but it's Euclidean geometry that is still valid,
0:50:54 > 0:50:58and even in higher mathematics, when you go to higher dimensional spaces,
0:50:58 > 0:51:00you're still using Euclidean geometry.
0:51:02 > 0:51:06Alexandria must have been an inspiring place for the ancient scholars,
0:51:06 > 0:51:12and Euclid's fame would have attracted even more eager, young intellectuals to the Egyptian port.
0:51:12 > 0:51:18One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes.
0:51:19 > 0:51:23He would become a mathematical visionary.
0:51:23 > 0:51:28The best Greek mathematicians, they were always pushing the limits,
0:51:28 > 0:51:29pushing the envelope.
0:51:29 > 0:51:32So, Archimedes...
0:51:32 > 0:51:35did what he could with polygons,
0:51:35 > 0:51:37with solids.
0:51:37 > 0:51:40He then moved on to centres of gravity.
0:51:40 > 0:51:44He then moved on to the spiral.
0:51:44 > 0:51:50This instinct to try and mathematise everything
0:51:50 > 0:51:54is something that I see as a legacy.
0:51:55 > 0:52:00One of Archimedes' specialities was weapons of mass destruction.
0:52:00 > 0:52:06They were used against the Romans when they invaded his home of Syracuse in 212 BC.
0:52:06 > 0:52:10He also designed mirrors, which harnessed the power of the sun,
0:52:10 > 0:52:12to set the Roman ships on fire.
0:52:12 > 0:52:17But to Archimedes, these endeavours were mere amusements in geometry.
0:52:17 > 0:52:20He had loftier ambitions.
0:52:23 > 0:52:29Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake
0:52:29 > 0:52:33and not for the ignoble trade of engineering or the sordid quest for profit.
0:52:33 > 0:52:37One of his finest investigations into pure mathematics
0:52:37 > 0:52:41was to produce formulas to calculate the areas of regular shapes.
0:52:43 > 0:52:49Archimedes' method was to capture new shapes by using shapes he already understood.
0:52:49 > 0:52:52So, for example, to calculate the area of a circle,
0:52:52 > 0:52:57he would enclose it inside a triangle, and then by doubling the number of sides on the triangle,
0:52:57 > 0:53:02the enclosing shape would get closer and closer to the circle.
0:53:02 > 0:53:04Indeed, we sometimes call a circle
0:53:04 > 0:53:07a polygon with an infinite number of sides.
0:53:07 > 0:53:11But by estimating the area of the circle, Archimedes is, in fact,
0:53:11 > 0:53:15getting a value for pi, the most important number in mathematics.
0:53:16 > 0:53:22However, it was calculating the volumes of solid objects where Archimedes excelled.
0:53:22 > 0:53:25He found a way to calculate the volume of a sphere
0:53:25 > 0:53:30by slicing it up and approximating each slice as a cylinder.
0:53:30 > 0:53:33He then added up the volumes of the slices
0:53:33 > 0:53:36to get an approximate value for the sphere.
0:53:36 > 0:53:39But his act of genius was to see what happens
0:53:39 > 0:53:42if you make the slices thinner and thinner.
0:53:42 > 0:53:47In the limit, the approximation becomes an exact calculation.
0:53:51 > 0:53:56But it was Archimedes' commitment to mathematics that would be his undoing.
0:53:58 > 0:54:02Archimedes was contemplating a problem about circles traced in the sand.
0:54:02 > 0:54:05When a Roman soldier accosted him,
0:54:05 > 0:54:11Archimedes was so engrossed in his problem that he insisted that he be allowed to finish his theorem.
0:54:11 > 0:54:16But the Roman soldier was not interested in Archimedes' problem and killed him on the spot.
0:54:16 > 0:54:21Even in death, Archimedes' devotion to mathematics was unwavering.
0:54:43 > 0:54:46By the middle of the 1st century BC,
0:54:46 > 0:54:50the Romans had tightened their grip on the old Greek empire.
0:54:50 > 0:54:53They were less smitten with the beauty of mathematics
0:54:53 > 0:54:56and were more concerned with its practical applications.
0:54:56 > 0:55:02This pragmatic attitude signalled the beginning of the end for the great library of Alexandria.
0:55:02 > 0:55:06But one mathematician was determined to keep the legacy of the Greeks alive.
0:55:06 > 0:55:11Hypatia was exceptional, a female mathematician,
0:55:11 > 0:55:14and a pagan in the piously Christian Roman empire.
0:55:16 > 0:55:21Hypatia was very prestigious and very influential in her time.
0:55:21 > 0:55:27She was a teacher with a lot of students, a lot of followers.
0:55:27 > 0:55:31She was politically influential in Alexandria.
0:55:31 > 0:55:34So it's this combination of...
0:55:34 > 0:55:40high knowledge and high prestige that may have made her
0:55:40 > 0:55:44a figure of hatred for...
0:55:44 > 0:55:46the Christian mob.
0:55:51 > 0:55:55One morning during Lent, Hypatia was dragged off her chariot
0:55:55 > 0:55:59by a zealous Christian mob and taken to a church.
0:55:59 > 0:56:03There, she was tortured and brutally murdered.
0:56:06 > 0:56:09The dramatic circumstances of her life and death
0:56:09 > 0:56:12fascinated later generations.
0:56:12 > 0:56:17Sadly, her cult status eclipsed her mathematical achievements.
0:56:17 > 0:56:20She was, in fact, a brilliant teacher and theorist,
0:56:20 > 0:56:26and her death dealt a final blow to the Greek mathematical heritage of Alexandria.
0:56:33 > 0:56:37My travels have taken me on a fascinating journey to uncover
0:56:37 > 0:56:42the passion and innovation of the world's earliest mathematicians.
0:56:42 > 0:56:47It's the breakthroughs made by those early pioneers of Egypt, Babylon and Greece
0:56:47 > 0:56:52that are the foundations on which my subject is built today.
0:56:52 > 0:56:55But this is just the beginning of my mathematical odyssey.
0:56:55 > 0:56:59The next leg of my journey lies east, in the depths of Asia,
0:56:59 > 0:57:02where mathematicians scaled even greater heights
0:57:02 > 0:57:04in pursuit of knowledge.
0:57:04 > 0:57:08With this new era came a new language of algebra and numbers,
0:57:08 > 0:57:12better suited to telling the next chapter in the story of maths.
0:57:12 > 0:57:16You can learn more about the story of maths
0:57:16 > 0:57:19with the Open University at...
0:57:36 > 0:57:39Subtitles by Red Bee Media Ltd